Hardware-efficient variational quantum eigenvalue solver for quantum computing machines

ABSTRACT

Generating trial states for a variational quantum Eigenvalue solver (VQE) using a quantum computer is described. An example method includes selecting a number of samples S to capture from qubits for a particular trial state. The method further includes mapping a Hamiltonian to the qubits according the trial state. The method further includes setting up an entangler in the quantum computer, the entangler defining an entangling interaction between a subset of the qubits of the quantum computer. The method further includes reading out qubit states after post-rotations associated with Pauli terms in the target Hamiltonian, the reading out being performed for S samples. The method further includes computing an energy state using the S qubit states. The method further includes, in response to the estimated energy state not converging with an expected energy state, computing a new trial state for the VQE and iterating to compute the estimated energy using the new trial state.

CROSS-REFERENCES TO RELATED APPLICATIONS

This patent application claims priority to U.S. Provisional PatentApplication Ser. No. 62/561,840, filed Sep. 22, 2017, which isincorporated herein by reference in its entirety.

STATEMENT REGARDING PRIOR DISCLOSURES BY THE INVENTOR OR A JOINTINVENTOR

The following disclosure(s) are submitted under 35 U.S.C. 102(b)(1)(A)as having been made by or having originated from one or more members ofthe inventive entity of the application:

“Hardware-efficient Quantum Optimizer for Small Molecules and QuantumMagnets”, Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, MaikaTakita, Jerry M. Chow, Jay M. Gambetta, 17 Apr. 2017 (arXiv:1704.05018).

“Hardware-efficient Variational Quantum Eigensolver for Small Moleculesand Quantum Magnets”, Abhinav Kandala, Antonio Mezzacapo, Kristan Temme,Maika Takita, Markus Brink, Jerry M. Chow, Jay M. Gambetta, 14 Sep. 2017(Nature, 549, 242-246 (2017)).

BACKGROUND

The present application relates in general to quantum computing, andmore specifically to implementation of a hardware-efficient variationalquantum eigenvalue solver for quantum computing machines.

Quantum computing uses particle physics, which defines a fermion as anyparticle characterized by Fermi-Dirac statistics. These particles obeythe Pauli Exclusion Principle. Fermions include all quarks and leptons,as well as any composite particle made of an odd number of these, suchas all baryons and many atoms and nuclei. Fermions differ from bosons,which obey Bose-Einstein statistics. A fermion can be an elementaryparticle, such as the electron, or it can be a composite particle, suchas the proton. According to the spin-statistics theorem in anyreasonable relativistic quantum field theory, particles with integerspin are bosons, while particles with half-integer spin are fermions.

In addition to a spin characteristic, fermions also possess conservedbaryon or lepton quantum numbers. Therefore, what is usually referred toas the spin statistics relation is in fact a spin statistics-quantumnumber relation. As a consequence of the Pauli Exclusion Principle, onlyone fermion can occupy a particular quantum state at any given time. Ifmultiple fermions have the same spatial probability distribution, atleast one property of each fermion, such as its spin, must be different.Fermions are usually associated with matter, whereas bosons aregenerally force carrier particles, although in the current state ofparticle physics the distinction between the two concepts is unclear.Weakly interacting fermions can also display bosonic behavior underextreme conditions. At low temperatures, fermions show superfluidity foruncharged particles and superconductivity for charged particles.Composite fermions, such as protons and neutrons, are the key buildingblocks of everyday matter. Quantum computing machines use suchcharacteristics of the particles to solve various computationallyexpensive problems.

Quantum computing has emerged based on its applications in, for example,cryptography, molecular modeling, materials science condensed matterphysics, and various other fields, which currently stretch the limits ofexisting high-performance computing resources for computational speedup.At the heart of a quantum computing machines lies the utilization ofqubits (i.e., quantum bits), whereby a qubit may, among other things, beconsidered the analogue of a classical bit (i.e., digital bit—‘0’ or‘1’) having two quantum mechanical states (e.g., a high state and a lowstate) such as the spin states of an electron (i.e., ‘1’=↑ and ‘0’=↓),the polarization states of a photon (i.e., ‘1’=H and ‘0’=V), or theground state (‘0’) and first excited state (‘1’) of a transmon, which isa superconducting resonator made from a capacitor in parallel with aJosephson junction acting as a non-linear inductor. Although qubits arecapable of storing classical ‘1’ and ‘0’ information, they also presentthe possibility of storing information as a superposition of ‘1’ and ‘0’states.

For quantum computing machines, where the dimension of the problem spacegrows exponentially, finding the eigenvalues of certain operators can bean intractable problem.

SUMMARY

According to one or more embodiments, an example system includes amemory device including computer-executable instructions. The systemfurther includes a processor coupled with the memory. The processorexecutes the computer-executable instructions to generate trial statesfor a variational quantum Eigenvalue solver (VQE) using a quantumcomputer that comprises a plurality of qubits. The generation of trialstates includes selecting a number of samples S to capture from thequbits for a particular trial state, the samples comprising measurementsof the qubit states. The generation of trial states further includesmapping a Hamiltonian to the qubits of the quantum computer according tothe trial state. The generation of trial states further includes settingup an entangler in the quantum computer, the entangler defining aninteraction between at least a subset of the qubits of the quantumcomputer. The generation of trial states further includes reading out,from the quantum computer, qubit states after post-rotations associatedwith Pauli terms, the reading out being performed for the selectednumber of samples S. The generation of trial states further includescomputing an estimated energy state using the S measurements of eachPauli term in the Hamiltonian. The generation of trial states furtherincludes, in response to the estimated energy state not converging withan expected energy state, computing a new trial state for the VQE anditerating to compute the estimated energy using the second trial state.

According to one or more embodiments, an example method for togenerating trial states for a variational quantum Eigenvalue solver(VQE) using a quantum computer that comprises a plurality of qubitsincludes selecting a number of samples S to capture from the qubits fora particular trial state, the samples being measurements of the qubitstates. The generation of trial states further includes mapping aHamiltonian to the qubits of the quantum computer. The generation oftrial states further includes setting up an entangler in the quantumcomputer, the entangler defining an entangling interaction between atleast a subset of the qubits of the quantum computer. The generation oftrial states further includes reading out, from the quantum computer,qubit states after post-rotations associated with Pauli terms in theHamiltonian, the reading out being performed for the selected number ofsamples S. The generation of trial states further includes computing anestimated energy state using the S measurements of each Pauli term inthe Hamiltonian. The generation of trial states further includes inresponse to the estimated energy state not converging with an expectedenergy state, computing a new trial state for the VQE and iterating tocompute the estimated energy using the new trial state.

According to one or more embodiments, an example quantum computingdevice includes multiple qubits. The quantum computing device furtherincludes a plurality of resonators corresponding to each of the qubits,each resonator receiving control signals for the corresponding qubit,and receiving a readout signal for measurement of the qubit state of thecorresponding qubit. The quantum computing device further includesresonators that couple qubits in the quantum device. The quantumcomputing device generates trial states for a variational quantumEigenvalue solver (VQE). The generation of trial states includesreceiving, by the resonators, control pulses associated with the trialstate. The generation of trial states further includes setting up anentangler according to the trial state, the entangler defining aninteraction between at least a subset of the qubits. The generation oftrial states further includes reading out, by the resonators, qubitstates after post-rotations associated with Pauli terms, the reading outbeing performed for a selected number of samples S for computing anestimated energy state. The generation of trial states further includes,in response to the estimated energy state not converging with anexpected energy state, receiving control signals to update the entangleraccording to a new trial state for the VQE.

Additional features and advantages are realized through the techniquesof the present invention. Other embodiments and aspects of the inventionare described in detail herein and are considered a part of the claimedinvention. For a better understanding of the invention with theadvantages and the features, refer to the description and to thedrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an example of a classical computer configured to execute thecomputations related to a Hamiltonian according to one or moreembodiments of the present invention; and

FIG. 2 is an example of a quantum computer configured to execute theoutput of the classical computer related to the Hamiltonian according toone or more embodiments of the present invention.

FIG. 3 depicts a block diagram of the quantum computer according to oneor more embodiments.

FIG. 4 depicts a quantum circuit for trial state preparation and energyestimation according to one or more embodiments.

FIG. 5 depicts an example pulse sequence according to one or moreembodiments.

FIG. 6 illustrates a flowchart of an example method for executing ahardware-efficient variational quantum eigenvalue solver using a quantumcomputing machine according to one or more embodiments.

FIG. 7 depicts example plots depicting calibrations for parameters in anexample scenario.

FIG. 8 depicts an example optimization of molecular structureapplication using the improved VQE according to one or more embodiments.

FIG. 9 depicts a comparison of results using the improved VQE using thetrial states according to one or more embodiments with known values.

DETAILED DESCRIPTION

In quantum mechanics, the Hamiltonian is the operator corresponding tothe total energy of the system in most of the cases. It is usuallydenoted by H, also H or H. Its spectrum is the set of possible outcomeswhen one measures the total energy of a system. Because of its closerelation to the time-evolution of a system, it is of fundamentalimportance in most formulations of quantum theory.

Quantum information processing holds the promise of solving specificcomputational problems, which are deemed to be too challenging forconventional classical hardware. A computational task which isparticularly well suited for quantum computers is the simulation ofquantum mechanical systems. The central application here is thesimulation of strongly interacting Fermionic systems, which can forinstance be found in quantum chemistry, materials science, and nuclearphysics. In order to represent Fermionic degrees of freedom on a quantumcomputer, the Fermionic modes need to be mapped to qubits that are theelementary logical units of quantum computation. It should be noted thatfermionic simulation is only one possible application of quantumcomputing, and other applications include quantum spin systems such asquantum magnets and combinatorial optimization problems.

It is known, however, that physical Fermionic systems obey parityconservation and in some cases the even stronger conservation ofparticle number. Hamiltonians or aspects of Hamiltonians are encodedonto qubits in the qubit simulation. The simulation is executed on aquantum computer having qubits. It is noted that a qubit is a physicalpiece of quantum hardware in a quantum computer and the qubit is asuperconducting quantum device. In a Hamiltonian, the qubit is used as aterm that represents the physical qubit.

The input to the quantum computer is the Hamiltonian (or terms of theHamiltonian) that is used to encode a system that is to be simulated.The output is energy measured from the quantum computer. A quantumalgorithm is a finite sequence of step-by-step instructions sent to thequantum computer to solve a particular problem. Here, experimenters areinterested in obtaining estimates for the ground state energy of theinput Hamiltonian. Therefore, this input Hamiltonian leads to a set ofinput data for a quantum simulation on the quantum computer.

A computer 100 in FIG. 1 is programmed and configured to execute theencoding of quantum Hamiltonians on a set of qubits and any other(preparation) computations that are not performed on the quantumcomputer. One or more software applications 160 are programmed withcomputer instructions, such that processors 110 can execute them. Afterexecuting the respective preparation computations on the computer 100,the output (i.e., the output Hamiltonian) is then applied and executedon a quantum computer 200 in FIG. 2. There can be a feedback loop (backand forth process) of performing some computations on the computer 100,which are then fed and used by the quantum computer 200. The quantumcomputer 200 is for exemplary purposes only and it not meant to be theexact structure of the quantum computer. As understood by one skilled inthe art, the classical computer 100 in FIG. 1 runs the determination ofpulses, etc., and the quantum computer 200 in FIG. 2 runs the output.

FIG. 2 is an example of a quantum computer 200 (quantum hardware) thatcan process the output from the computer 100 according to embodiments ofthe present invention. A quantum processor is a computing device thatcan harness quantum physical phenomena (such as superposition,entanglement, and quantum tunneling) unavailable to non-quantum devices.A quantum processor may take the form of a superconducting quantumprocessor. A superconducting quantum processor may include a number ofqubits and associated local bias devices, for instance two or moresuperconducting qubits. An example of a qubit is a flux qubit. Asuperconducting quantum processor may also employ coupling devices(i.e., “couplers”) providing communicative coupling between qubits.

The number of qubits, the interaction of qubits, and the configurationof qubits are all meant for example purposes and not limitation. Itshould be appreciated that the qubits (and readout resonators which arenot shown) can be constructed in various different configuration andFIG. 2 is not meant to be limiting. In general, a quantum computer isany physical system that obeys the laws of quantum mechanics whichsatisfy the DiVincenzo criteria. These criteria set the requirements onthe quantum mechanical system to be considered a quantum computer. Thecriteria include (1) a scalable physical system with well-characterizedqubits, (2) the ability to initialize the state of the qubits to asimple fiducial state, (3) long relevant decoherence times, (4) a“universal” set of quantum gates, (5) a qubit-specific measurementcapability, (6) the ability to interconvert stationary and flyingqubits, and (7) the ability to faithfully transmit flying qubits betweenspecified locations.

The quantum computer 200 in FIG. 2 illustrates an input 205 as a controlprogram, control signals 210, qubits 215, readout signals 220, andmeasurement data 225 as the output. As a quantum mechanical system thatsatisfies these requirements, the quantum computer 200 is configured toreceive control signals 210 as input 205 information (e.g., according toterms, aspects, etc., of the Hamiltonian) to apply a sequence of quantumgates and apply measurement operations. The quantum gates betweendifferent qubits 215 are mediated through their interactions 230. Themeasurement operators produce classical signals (as measurement data225) that can be read by an experimenter controlling the system, i.e.,the quantum computer 200.

Now turning back to FIG. 1, an example illustrates a computer 100, e.g.,any type of computer system configured to execute algorithm(s)(including various mathematical computation as understood by one skilledin the art) for encoding quantum Hamiltonians on a set of qubits, asdiscussed herein, such that the result can be input to the quantumcomputer 200. The computer 100 can be a distributed computer system overmore than one computer. Various methods, procedures, modules, flowdiagrams, tools, applications, circuits, elements, and techniquesdiscussed herein can also incorporate and/or utilize the capabilities ofthe computer 100. Indeed, capabilities of the computer 100 can beutilized to implement elements of exemplary embodiments discussedherein.

Generally, in terms of hardware architecture, the computer 100 caninclude one or more processors 110, computer readable storage memory120, and one or more input and/or output (I/O) devices 170 that arecommunicatively coupled via a local interface (not shown). The localinterface can be, for example but not limited to, one or more buses orother wired or wireless connections, as is known in the art. The localinterface can have additional elements, such as controllers, buffers(caches), drivers, repeaters, and receivers, to enable communications.Further, the local interface can include address, control, and/or dataconnections to enable appropriate communications among theaforementioned components.

The processor 110 is a hardware device for executing software that canbe stored in the memory 120. The processor 110 can be virtually anycustom made or commercially available processor, a central processingunit (CPU), a data signal processor (DSP), or an auxiliary processoramong several processors associated with the computer 100, and theprocessor 110 can be a semiconductor based microprocessor (in the formof a microchip) or a macroprocessor.

The computer readable memory 120 can include any one or combination ofvolatile memory elements (e.g., random access memory (RAM), such asdynamic random access memory (DRAM), static random access memory (SRAM),etc.) and nonvolatile memory elements (e.g., ROM, erasable programmableread only memory (EPROM), electronically erasable programmable read onlymemory (EEPROM), programmable read only memory (PROM), tape, compactdisc read only memory (CD-ROM), disk, diskette, cartridge, cassette orthe like, etc.). Moreover, the memory 120 can incorporate electronic,magnetic, optical, and/or other types of storage media. Note that thememory 120 can have a distributed architecture, where various componentsare situated remote from one another, but can be accessed by theprocessor(s) 110.

The software in the computer readable memory 120 can include one or moreseparate programs, each of which includes an ordered listing ofexecutable instructions for implementing logical functions. The softwarein the memory 120 includes a suitable operating system (O/S) 150,compiler 140, source code 130, and one or more applications 160 of theexemplary embodiments. As illustrated, the application 160 includesnumerous functional components for implementing the elements, processes,methods, functions, and operations of the exemplary embodiments.

The operating system 150 can control the execution of other computerprograms, and provides scheduling, input-output control, file and datamanagement, memory management, and communication control and relatedservices.

The application 160 can be a source program, executable program (objectcode), script, or any other entity comprising a set of instructions tobe performed. When a source program, then the program is usuallytranslated via a compiler (such as the compiler 140), assembler,interpreter, or the like, which can be included within the memory 120,so as to operate properly in connection with the O/S 150. Furthermore,the application 160 can be written as (a) an object oriented programminglanguage, which has classes of data and methods, or (b) a procedureprogramming language, which has routines, subroutines, and/or functions.

The I/O devices 170 can include input devices (or peripherals) such as,for example but not limited to, a mouse, keyboard, scanner, microphone,camera, etc. Furthermore, the I/O devices 170 can also include outputdevices (or peripherals), for example but not limited to, a printer,display, etc. Finally, the I/O devices 170 can further include devicesthat communicate both inputs and outputs, for instance but not limitedto, a NIC or modulator/demodulator (for accessing remote devices, otherfiles, devices, systems, or a network), a radio frequency (RF) or othertransceiver, a telephonic interface, a bridge, a router, etc. The I/Odevices 170 also include components for communicating over variousnetworks, such as the Internet or an intranet. The I/O devices 170 canbe connected to and/or communicate with the processor 110 utilizingBluetooth connections and cables (via, e.g., Universal Serial Bus (USB)ports, serial ports, parallel ports, FireWire, HDMI (High-DefinitionMultimedia Interface), etc.).

In exemplary embodiments, where the application 160 is implemented inhardware, the application 160 can be implemented with any one or acombination of the following technologies, which are each well known inthe art: a discrete logic circuit(s) having logic gates for implementinglogic functions upon data signals, an application specific integratedcircuit (ASIC) having appropriate combinational logic gates, aprogrammable gate array(s) (PGA), a field programmable gate array(FPGA), etc.

The technical solutions provided by the present invention addressestechnical challenges for quantum computing machines when the dimensionof a problem space grows exponentially, whereby finding the eigenvaluesof certain operators can be an intractable problem. For example,consider electronic-structure problems, in which the fundamental goal isto determine state of motion of electrons in an electrostatic fieldcreated by stationary nuclei, and which includes solving for theground-state energy of many-body interacting fermionic Hamiltonians.Solving such problems on a quantum computer relies on a mapping betweenfermionic and qubit operators, which restates the problem as a specificinstance of a local Hamiltonian problem on a set of qubits. Given ak-local Hamiltonian H, composed of terms that act on at most k qubits,the solution to the local Hamiltonian problem amounts to finding itsground-state eigenvalue E_(G) and ground state |Φ_(G)>, which satisfyH|Φ_(G)>=E_(G)|Φ_(G)>. For k≥2, the problem is known to be QuantumMerlin Arthur (QMA)-complete. It should be noted that the technicalsolutions described herein are applicable to various other technicalproblems other than the electronic structure problem, such as quantummagnetism and the like.

In one or more examples, the technical solutions for addressing thetechnical problems, such as the electronic structure problems and thelike, include using a variational quantum eigenvalue solver (VQE), whichis used to find the lowest energy state of a target Hamiltonian. In oneor more examples, the implementation of the VQE using the quantumcomputer 200 includes using experimentally accessible controlparameters, to prepare trial states for the quantum computer 200, andmeasuring the energy associated with the prepared trial states. Themeasured energy is then fed to an optimization routine to generate anext set of control parameters that tend to lower the energy. Iterationsare performed until the lowest energy is obtained to the desiredaccuracy.

Technical challenges of implementing the above protocol using trialstates such as the unitary couple cluster (UCC) ansatz prepared quantumhardware, include restrictions due to coherence and controls, amongothers. For example, the UCC ansatz has a number of variationalparameters that scale quartically with the number of spin orbitals thatare considered in the single- and double-excitation approximation.Further, implementation of the UCC ansatz is sensitive to trotter andgate errors. Other technical challenges with implementations of VQEinclude the requirement of a large number of samples, which can resultin an unreasonably large experiment time in the absence of resetschemes. Also, the number of measurements required for application ofVQE to electronic structure problems is equal to the number of terms inthe Hamiltonian, which in the most general case, scales as ˜N⁴ forelectronic structure problems, where N is the number of spin orbitals.This can further lead to large experiment time.

The technical solutions described herein address such technicalchallenges of using the quantum computer 200 to implement the VQE byfacilitating hardware efficient trial state preparation suited toimplementations with limited coherence of existing quantum hardware.Further, the technical solutions facilitate implementation(s) of a VQEthat is insensitive to coherent gate errors. In one or more examples,the technical solutions described herein use a fixed frequencyarchitecture with all-microwave qubit control, which maximizes quantumcoherence of the quantum hardware, in the absence of frequency tunablecomponents that are susceptible to flux noise, etc. Further, by using amicrowave-only single qubit reset, the technical solutions describedherein facilitate maintaining qubit coherence, while reducing totaloptimization time. The technical solutions described herein furtherfacilitate parallelization of gates thus reducing trial statepreparation time. In addition, the technical solutions described hereinfacilitate grouping of Hamiltonian terms measured by a unique set ofsingle qubit rotation, which facilitates reducing a total number ofmeasurements to be captured and analyzed, thus reducing the total timeof the simulation. It should be noted that the technical solutionsdescribed herein provide additional advantages and benefits than thoselisted above, as will be evident to a person skilled in the art.

The technical solutions described herein facilitate implementing ahardware-efficient UCC ansatz preparation for a VQE, whereby trialstates are parameterized by quantum gates that are tailored to thephysical quantum computer 200 that is available.

FIG. 3 depicts a block diagram of the quantum computer according to oneor more embodiments. The structure of the quantum computer 300 asdepicted in FIG. 3 and described herein is for explanation of one ormore embodiments, and not limiting the technical solutions describedherein. As depicted, the quantum computer 300 includes seven qubits 215,such as Josephson junction (JJ) based transmon qubits, of which six arefixed frequency transmon qubits, and one is a central flux-tunableasymmetric transmon qubit. In other examples, the quantum hardware mayinclude different number of qubits and a different combination offixed-frequency and flux-tunable qubits.

Each qubit 215 has its own individual coplanar waveguide (CPW) resonator216 for receiving control signals 210 and receiving readout signals 220.The qubits 215 are readout by dispersive measurements through theindependent readout resonators 216, with each readout line having asequence of low temperature amplifiers 310—a Josephson parametricconverter (JPC1-JPC6) followed by a high electron mobility transistor320 (HEMT; M1-M6) for achieving high assignment fidelity.

In one or more examples, the qubits are controlled solely by microwavepulses that are delivered via attenuated coaxial lines. The single qubitgates are implemented by microwave drives at the specific qubit Q_(i)'sfrequency f_(i), while the entangling two-qubit CR gates are implementedby driving a control qubit Qc at the frequency ω_(t) of the target qubitQ_(t), where i, c, t∈{1; 2; 3; 4; 5; 6}, in the depicted example with 6or more qubits. It should be noted that in other examples that usequantum hardware with different number of qubits, the qubit coupling mayuse different number of control qubits and/or target qubits.

Further, the state of each qubit 215 is measured at its readoutresonator frequency ω_(Mi). The reflected readout signals are amplifiedfirst by the JPC 310, pumped at a frequency ω_(Pi), followed by the HEMT320 amplifiers, for example at 4K. Each qubit 215 has correspondingcharacteristics, the characteristics including qubit transitions(ω₀₁/2π), average relaxation times (T₁), average coherence times (T₂,T₂*), readout resonator frequencies (ω_(r)/2π), qubit anharmonicity(δ/2π), readout assignment errors (ε_(r)), among others.

Further, to maintain stability of the gates used for trial statepreparation, even during the long times associated with optimization oflarge Hamiltonians, the amplitude and phase of the single-qubit andtwo-qubit gates are calibrated during the course of usage. For example,to estimate the time scale and magnitude of drifts in pulse amplitudeand phase, the gates are calibrated periodically over a predeterminedduration, such as 6 hours, 10 hours, or any other duration.

Further, the quantum computer 300 includes shared CPW resonators 218 forqubit-qubit coupling, which are referred to as “quantum buses” 218. Itshould be noted that although FIG. 3 depicts two resonators providingthe qubit-qubit coupling, in other examples, more resonators may be usedto provide such coupling. Further, as depicted in FIG. 3, the quantumbuses 218 enable one or more two-qubit entangling gates, such asmicrowave-only cross resonance (CR) gates which may be used to implementthe entangler U_(ENT) 402. In one or more examples, the entangler mayinclude a different implementation, such as single-bit gates, or thelike.

In one or more examples, the gates that constitute the entanglersU_(ENT) in the trial state preparation are implemented by driving acontrol qubit Q_(c) with a microwave pulse that is resonant with atarget qubit Q_(t). With the addition of single qubit rotations, the CRgate can be used to construct a controlled NOT (CNOT). In one or moreexamples, the entangling gate phase may be an additional variationalparameter for the optimization of the VQE.

FIG. 4 depicts a quantum circuit for trial state preparation and energyestimation according to one or more embodiments. The depicted examplecircuit is for 6 qubits, however in other examples, the quantum circuitcan use a different number of qubits, fewer, or more than six. Thecircuit 400 is composed of a sequence of interleaved single-qubit 215rotations 401, and entangling unitary operations U_(ENT) 402 thatentangle all the qubits 215 in the circuit 400.

An entangler 402 is a sequence of gates that entangle one or more of thequbits 215. Entanglement is delivered control pulses that are providedto the qubits 215. For example, an entangling interactions of thesuperconducting hardware of the quantum computer 300 may be described bya drift Hamiltonian H₀ that generates the entanglersU_(ENT)=exp(−iH_(0τ)). For the hardware efficient trial states, theseare interleaved with arbitrary single-qubit Euler rotations 401 whichare implemented as a combination of Z and X gates, given byU^(q,i)({right arrow over (θ)})=Z_(θ) ₁ _(q,i) ^(q), X_(θ) ₂ _(q,i)^(q), Z_(θ) ₃ _(q,i) ^(q), where q identifies the qubit 215 and i=0, 1,. . . d refers to the depth position, as depicted in FIG. 4.

The Q-qubit trial states are obtained from the state |00 . . . 0>,applying d entanglers U_(ENT) that alternate with N Euler rotations,giving

$\left. {\Phi\left( \overset{\rightarrow}{\theta} \right)} \right\rangle = {\prod\limits_{q = 1}^{N}{\left\lbrack {U^{q,d}\left( \overset{\rightarrow}{\theta} \right)} \right\rbrack \times U_{ENT} \times {\prod\limits_{q = 1}^{N}{\left\lbrack {U^{q,{d - 1}}\left( \overset{\rightarrow}{\theta} \right)} \right\rbrack\ldots \times U_{ENT} \times {\prod\limits_{q = 1}^{N}{\left\lbrack {U^{q,0}\left( \overset{\rightarrow}{\theta} \right)} \right\rbrack{\left. {00\ldots\mspace{14mu} 0} \right\rangle.}}}}}}}$

Because the qubits 215 are initialized in their ground state |0>, thefirst set of Z rotations of U^(q,i)({right arrow over (θ)}) is notimplemented, resulting in a total of p=N(3d+2) independent angles. Itshould be noted that the evolution time τ and the individual couplingsin H₀ can be controlled. However, leaving the p control angles asvariational parameters in the trial states facilitates the technicalsolutions to determine accurate optimizations using fixed-phase U_(ENT).The hardware-efficient approach used by the technical solutionsdescribed herein does not rely on the accurate implementation ofspecific two qubit gates and can be used with any U_(ENT) that generatessufficient entanglement. This is an improvement to UCC trial states thatrequire high-fidelity quantum gates approximating a unitary operatortailored on a theoretical ansatz.

FIG. 5 depicts an example pulse sequence according to one or moreembodiments. The example depicts a pulse sequence 500 for thepreparation of a six qubit trial state for the quantum computer 300. Inone or more examples, the Z rotations are implemented as frame changesin the control software, while the X rotations are implemented byappropriately scaling the amplitude of calibrated Xπ pulses, using afixed total time of a predetermined durations, such as 100 ns, for everysingle-qubit rotation.

Further, the U_(ENT) 402 is implemented as a composition or sequence oftwo-qubit cross resonance gates. In one or more examples, the CRresonance gates are implemented as CRc-t gates, implemented by driving acontrol qubit Qc with a microwave pulse resonant with a target qubit Qt.Further, Hamiltonian tomography of the CRc-t gates is used to reveal thestrengths of the various interaction terms, and the gate time formaximal entanglement. Further, in one or more examples, the two-qubitgate times (τ) are setup at a fixed value, such as 150 ns, thatrepresents plateaus of minimal energy error around gate phasescorresponding to the maximal pairwise concurrence. Using a fixed gatetime further improves the performance of the quantum circuit 400 byminimizing the effect of decoherence without compromising the accuracyof the optimization outcome.

Referring back to FIG. 4, in one or more examples, after each trialstate is prepared, the associated energy is estimated by measuring theexpectation values of the individual Pauli terms in the Hamiltonian.These estimates are affected by stochastic fluctuations due to finitesampling effects.

Further, in one or more examples, different post-rotations 410 areapplied after trial state preparation for sampling different Paulioperators in the input Hamultonian. The states of the qubit is thenmeasured using read-out pulses 420. In one or more examples, the Paulioperators are grouped into tensor product basis sets that use the samepost-rotations 410. Such grouping reduces the energy fluctuations,keeping the same total number of samples, reducing in this way the timeover-head for energy estimation.

The energy estimates are then used, in one or more examples, by agradient descent algorithm that relies on a simultaneous perturbationstochastic approximation (SPSA) to update the control parameters. TheSPSA algorithm approximates the gradient using only two energymeasurements, regardless of the dimensions of the parameter space p,achieving a level of accuracy comparable to standard gradient descentmethods, in the presence of stochastic fluctuations. The technicalsolutions thus facilitate optimizing over multiple qubits and longdepths for trial state preparation, thus facilitating optimizations overa number of parameters, for example p=30. It should be noted that theSPSA is just an example optimizer and that in other examples, thetechnical solutions may be implemented using a different optimizationalgorithm.

In one or more examples, the optimizer algorithm is executed by thecomputer 100 using the energy states output by the quantum computer 300.The computer 100 updates the control parameters and thus generates anext trial state that is fed into the quantum computer 300, for furtherimprovement of the trial state. The back and forth process continuesuntil a convergence of the estimated energy, or if the resources of thequantum computer 300 are saturated (for example, decoherence limit,sampling time).

FIG. 6 illustrates a flowchart of an example method for executing avariational quantum eigenvalue solver using a quantum computing machineaccording to one or more embodiments. The flowchart is illustrated usingan example implementation of the method in an electronic structurefield, for determining the ground state energy for molecules, such asH₂, LiH, and BeH2 molecules. However, it is understood that the methodis applicable to any other molecules, and further to any other technicalproblem to which the quantum computing machine is applicable, such asquantum magnetism, or any other.

The method includes mapping a quantum Hamiltonian problem to a qubitHamiltonian H, as shown at 610. For example, for the molecular energyestimation, the molecular Hamiltonians considered are computed in theSTO-3G basis, using software such as PYQUANTE™. It is understood thatthe molecular quantum Hamiltonian can be computed in any way. Mappingthe Hamiltonian to the qubit includes determining a number ofspin-orbitals for the qubits 215 based on the computed Hamiltonian. Forexample, in the molecular Hamiltonian, for an H2 molecule, each atomcontributes a 1s orbital, for a total of 4 spin-orbitals. For example,the X axis is set as the interatomic axis for the LiH and BeH2molecules, and consider the orbitals is for each H atom and 1s, 2s,2p_(x) for the Li and Be atoms, assuming zero filling for the 2p_(y) and2p_(z) orbitals, which do not interact strongly with the subset oforbitals considered. This choice of orbitals amounts to a total of 8spin-orbitals for LiH and 10 for BeH2. It should be noted that in otherexamples, the mapping be different from the above description.

The Hamiltonians are expressed using the second quantization languageas,

${H = {{H_{1} + H_{2}} = {{\sum\limits_{\alpha,{\beta = 1}}^{M}{t_{\alpha\beta}a_{\alpha}^{\dagger}a_{\beta}}} + {\frac{1}{2}{\sum\limits_{\alpha,\beta,\gamma,{\delta = 1}}^{M}{u_{\alpha\beta\gamma\delta}a_{\alpha}^{\dagger}a_{\gamma}^{\dagger}a_{\delta}a_{\beta}}}}}}},$where, α_(α) ^(†) (α_(α)) is the fermionic creation(annihilation)operator of the fermionic mode α, satisfying fermionic commutation rules{α_(α), α_(β)}=0, {α_(α) ^(†), α_(β) ^(†)}=0, {α_(α), α_(β)^(†)}=δ_(αβ). Here, M=4; 8; 10 is the number of spin-orbitals for H2,LiH, and BeH2 respectively, and using chemists' notation for thetwo-body integrals,

${t_{\alpha\beta} = {\int{d\;{\overset{\rightarrow}{x}}_{1}{\Psi_{\alpha}\left( {\overset{\rightarrow}{x}}_{1} \right)}\left( {{- \frac{{\overset{\rightarrow}{\nabla}}_{1}^{2}}{2}} + {\sum\limits_{i}\frac{Z_{i}}{{\overset{\rightarrow}{r}}_{1\; i}}}} \right){\Psi_{\beta}\left( {\overset{\rightarrow}{x}}_{1} \right)}}}},{u_{\alpha\beta\gamma\delta} = {\int{\int{d\;{\overset{\rightarrow}{x}}_{1}d\;{\overset{\rightarrow}{x}}_{2}{\Psi_{\alpha}^{*}\left( {\overset{\rightarrow}{x}}_{1} \right)}{\Psi_{\beta}\left( {\overset{\rightarrow}{x}}_{1} \right)}\frac{1}{{\overset{\rightarrow}{r}}_{12}}{\Psi_{\gamma}^{*}\left( {\overset{\rightarrow}{x}}_{2} \right)}{\Psi_{\delta}\left( {\overset{\rightarrow}{x}}_{2} \right)}}}}},$where Zi are the defined nuclei charges, {right arrow over (r)}_(1i) and{right arrow over (r)}₁₂ are nuclei-electron and electron-electronseparations, Ψ_(α)({right arrow over (x)}₁) is the α-th orbital wavefunction, and it is assumed that the spin is conserved in thespin-orbital indices α, β, and α, β, γ, δ. Further, in the case of LiHand BeH2, the mapping considers perfect filling for the inner isorbitals, dressed in the basis in which H1 is diagonal. To this extent,the computer 100 may implement a Bogoliubov transformation on the modesα_(α)′=Σ_(β)U_(αβ)α_(β) such that

${H_{1}^{d} = {U^{\dagger}H_{1}U}},{H_{1}^{d} = {\sum\limits_{\alpha = 1}^{M}{\omega_{\alpha}^{\prime}a_{\alpha}^{\dagger}{a_{\alpha}^{\prime}.}}}}$

The “dressed” is modes of Li and Be are now to be filled for efficientlyobtaining an effective Hamiltonian acting on generic states of the form|Ψ>=α_(1s↑)′^(†)α_(1s↓)′^(†)(Σ_(β≠1sσ)ψ_(β)α_(β)′^(†))|0> where ψ_(β)are generic normalized coefficients, and 1sσ={1s↑,1s↓} refers to theinner is orbitals of Li and Be. Note that this approximation is validwhenever −ω_(1sσ)′>>|u_(αβγδ)′|∀σ, α, β, γ, δ, i.e. in the case of verylow-energy orbitals that do not interact strongly with the higher-energyones. The ansatz |Ψ

=α′_(1s↑) ^(†)α′_(1s↓) ^(†)(Σ_(β≠1sσ)ψ_(β)α′_(β) ^(†))|0

, allows to define an effective screened Hamiltonian on the is orbitalsfor the hydrogen atoms, and 2s and 2p_(x) for Lithium and Beryllium, fora total of 6 and 8 spin-orbitals for LiH and BeH2, respectively.According to this ansatz, the one-body fermionic terms containing thefilled orbitals will now contribute as a shift to the total energy (Ihere is the identity operator)ω_(1↑)′α_(1↑) ^(†)′α_(1↑)′→ω_(1↑) ′I,ω _(1↓)′α_(1↓) ^(†)′α_(1↓)′→ω_(1↓)′I,while some of the two-body interactions, containing the set F of 1sfilled modes of Li and Be, F={1s ↑, 1s ↓}, become effective one-body orenergy shift terms,

${\frac{u_{\alpha\beta\gamma\delta}^{\prime}}{2}a_{\alpha}^{\prime\dagger}a_{\gamma}^{\prime\dagger}a_{\delta}^{\prime\dagger}a_{\beta}^{\prime\dagger}}->\left\{ {\begin{matrix}{{\frac{u_{\alpha\beta\gamma\delta}^{\prime}}{2}a_{\gamma}^{\prime\dagger}a_{\delta}^{\prime\dagger}},} & {{\alpha = \beta},{\alpha \in F},{\left\{ {\gamma,\delta} \right\} \notin F}} \\{{\frac{u_{\alpha\beta\gamma\delta}^{\prime}}{2}a_{\alpha}^{\prime\dagger}a_{\beta}^{\prime\dagger}},} & {{\gamma = \delta},{\gamma \in F},{\left\{ {\alpha,\beta} \right\} \notin F}} \\{{{- \frac{u_{\alpha\beta\gamma\delta}^{\prime}}{2}}a_{\gamma}^{\prime\dagger}a_{\beta}^{\prime\dagger}},} & {{\alpha = \delta},{\alpha \in F},{\left\{ {\beta,\gamma} \right\} \notin F}} \\{{{- \frac{u_{\alpha\beta\gamma\delta}^{\prime}}{2}}a_{\alpha}^{\prime\dagger}a_{\delta}^{\prime\dagger}},} & {{\gamma = \beta},{\gamma \in F},{\left\{ {\alpha,\delta} \right\} \notin F}} \\{{\frac{u_{\alpha\beta\gamma\delta}^{\prime}}{2}I},} & {{\alpha = \beta},{\gamma = \delta},{\alpha \neq \gamma},{\left\{ {\alpha,\gamma} \right\} \in F}} \\{{{- \frac{u_{\alpha\beta\gamma\delta}^{\prime}}{2}}I},} & {{\alpha = \delta},{\gamma = \beta},{\alpha \neq \gamma},{\left\{ {\alpha,\gamma} \right\} \in F}}\end{matrix},} \right.$while the two-body operators containing an odd number of modes in F willbe neglected.

The fermionic Hamiltonian H=Σ_(α,β≠1sσ)t_(αβ)α′_(α)^(†)α′_(β)+½Σ_(α,β,γ,δ≠1sσ)u′_(αβγδ)α′_(α) ^(†)α′_(γ) ^(†)α′_(δ)α′_(β)obtained in this way is mapped to the qubits 215. It should be notedthat this is one specific reduction of the Hamiltonian used for the BeH2and LiH molecules, and that in other examples, the Hamiltonian can berepresented differently based on different reduction techniques. In oneor more examples, the H2 Hamiltonian is mapped onto 4 qubits using abinary-tree mapping. Further, the M spin-orbitals are ordered by listingfirst the M/2 spin-up ones and then the M/2 spin-down ones. When usingthe binary-tree mapping, this produces a qubit Hamiltonian diagonal inthe second and fourth qubit, which has the total particle and spin

₂ symmetries encoded in those qubits. In one or more examples, for theLiH and BeH2 Hamiltonians parity mapping is used, which has the two

₂ symmetries encoded in the M/2-th and M-th mode, even if the totalnumber of spin orbitals is not a power of 2, as in the case of H2.Further, the Z Pauli operators of the M/2-th and M-th qubits areassigned a value based on the total number of electrons m in the systemaccording to

$\left\{ {Z_{M/2},Z_{M}} \right\} = \left\{ {\begin{matrix}{\left\{ {{+ 1},{+ 1}} \right\},} & {{{mod}\left( {m,4} \right)} = 0} \\{\left\{ {{\pm 1},{- 1}} \right\},} & {{{mod}\left( {m,4} \right)} = 1} \\{\left\{ {{- 1},{+ 1}} \right\},} & {{{mod}\left( {m,4} \right)} = 2} \\{\left\{ {{\pm 1},{- 1}} \right\},} & {{{mod}\left( {m,4} \right)} = 3}\end{matrix},} \right.$where the +1(−1) on Z_(M) for even(odd) m implies an even(odd) totalelectron parity. The values +1, −1 and ±1 for Z_(M/2) mean that thetotal number of electrons with spin-up in the ground state is even, odd,or there is an even/odd degeneracy, respectively. In the last case both+1 and −1 can be used equivalently for Z_(M/2). The final qubit-taperedHamiltonians consist of 4, 99 and 164 Pauli terms supported on 2, 4, 6qubits, each having 2, 25 and 44 tensor product basis (TPB) sets for H2,LiH and BeH2, respectively. As noted earlier, the Hamiltonians aremapped to the qubits differently in other examples.

The Hamiltonian H is mapped to the qubits by using a parity mapping (forexample, see FIG. 8 805). The mapped Hamiltonian is a sum of Pauliterms. A measurement sample of a collection of Pauli terms in a TPB canbe measured by rotating in the TPB by using microwave pulses, and in oneor more examples, solely using microwave pulses. For example, themicrowave pulses are parameterized by variable amplitude and phase torepresent the Pauli operators/terms. Alternatively, in one or moreexamples, the microwave pulses are parameterized by a constant amplitudewith varying phase to represent the Pauli operators/terms.

Referring back to the flowchart of FIG. 6, the method further includesselecting depth d for quantum circuit 400, a number of qubit states S tomeasure for each TBP set, and a maximal number of control updates N, asshown at block 620. The number of samples S refers to a number ofrepetitions of a given measurement using a given trial state. The trialstate is an interleaved sequence of single qubit gates that performarbitrary rotations and entanglers. The entangler is the sequence ofgates that entangle all the qubits utilized in the optimization. Quantumcircuit depth d is the number of entanglers used in the trial statepreparation. The gate is a sequence of microwave pulses that areparameterized by waveform, amplitude, and time. For example, themicrowave pulses that can be parametrized by varying amplitude andphase. Alternatively, the microwave pulses can be parameterized byconstant amplitude with varying phase. For example, for representing atrial state, the amplitude and/or phase of the control signals sent viathe microwave pulses are configured to specific corresponding values.

Further, the method includes, for each TBP set, drawing S samples ofqubit states of the mapped Hamiltonians using the quantum computer 300by reading out the states of the qubits 215 that are setup according tothe trial state, as shown at block 630. Determining the qubit states viathe quantum computer 300 includes initializing the qubits 215 to groundstate, as shown at block 631. The computer 100 sends reset signals tothe quantum computer for such initialization. Further, the computer 100sends control signals 210 to drive the qubits 215 to respective initialtrial states, as shown at block 632. Further, the quantum computer 300drives the qubits 215 to respective trial states during the iterations.

In one or more examples, the qubits are set to the trial statesemploying parallelization of gates to reduce trial state preparationtime. For example, the qubits 215 are setup concurrently with thecorresponding pulse sequences 500 representing the trial state.

Further, single-qubit post-rotations 410 associated with Pauli terms inthe TBP set are performed, as shown at block 633. The post-rotationsfacilitates the readout signals 220 from the qubits 215 to compute theexpectation value of the Hamiltonian operator(s) mapped onto the qubits215. Once mapped to the qubits 215, as described earlier, everymolecular Hamiltonian is expressed as a weighted sum of T Pauli termssupported on Q qubits H=Σ_(α=1) ^(T)h_(α)P_(α), where each P_(α)∈ {X, Y,Z, I}^(⊗Q) is a tensor product of single-qubit Pauli operators X, Y, Z,and the identity I, on Q qubits, with h_(α) being real coefficients.

The energy state estimate is given by

Φ({right arrow over (θ)}_(k))|H|Φ({right arrow over (θ)}_(k))

=

H

_(k) for the k-th control update. This can be done by averagingmeasurements outcomes from each iteration, using the same initial state,applying the quantum gates parametrized by {right arrow over (θ)}_(k),and finally performing projective measurements on the individual qubits.Because access to direct measurements of the Hamiltonian operator

H

and its variance is not available, the individual Pauli operators P_(α)are sampled and the mean values and variances

P_(α)

,

P_(α) ²

,=

P_(α) ²−

P_(α) ²

²

are estimated from the measurements outcomes of the α-th Pauli operator.The energy and Hamiltonian variance can then be obtained as

${\left\langle H \right\rangle = {\sum\limits_{\alpha = 1}^{T}{h_{\alpha}\left\langle P_{\alpha} \right\rangle}}},{{{Var}\lbrack H\rbrack} = {\sum\limits_{\alpha = 1}^{T}{h_{\alpha}^{2}\left\langle {\Delta\; P_{\alpha}^{2}} \right\rangle}}}$

The variance on the mean energy Var[H] is different from

ΔH²

because the individual Pauli terms are sampled separately; for example,eigenstates of H have

ΔH²

=0, but a finite Var[H]≠0. The error on the mean

$\epsilon = {\sqrt{\frac{{Var}\lbrack H\rbrack}{S}} \leq \sqrt{\frac{T{h_{{ma}\; x}^{2}}}{S}}}$where h_(max)=max_(α)|h_(α)| is the absolute value of the largest Paulicoefficient. Because sampling S times for a large number of trial statesand Pauli operators can cause significant time overhead, using the samestate preparations to measure different Pauli operators facilitatessaving such overhead for collecting multiple samples.

Hence, in one or more examples, the quantum computer 300 groupsdifferent Pauli terms for improving time efficiency. The individualPauli operators are measured by correlating measurement outcomes ofsingle-qubit dispersive readouts in the Z basis, which can be donesimultaneously because each qubit 215 is provided with an individualreadout resonator 216. In case a target multi-qubit Pauli operatorcontains non-diagonal single-qubit Pauli operator, single-qubitrotations (post-rotations) are performed before the measurement in the Zbasis. For example, specifically, a −π/2(n/2) rotation along the X(Y)axis to measure a Y(X) single-qubit Pauli operator.

Further, To minimize sampling overheads, we group the T Pauli operatorsP_(α) in A sets s₁, s₂, . . . s_(A), which have terms that are diagonalin the same tensor product basis. The post-rotations required to measureall the Pauli terms in a given TPB set are the same, and a unique statepreparation can be used to sample all the Pauli operators in the sameset. By doing so, however, covariance effects in the same TPB setcontribute to the variance of the total Hamiltonian,

${{{Var}^{G}\lbrack H\rbrack} = {{\sum\limits_{i = 1}^{A}{\sum\limits_{\alpha,{\beta \in s_{i}}}{h_{\alpha}h_{\beta}\left\langle {\left( {P_{\alpha} - \left\langle P_{\alpha} \right\rangle} \right)\left( {P_{\beta} - \left\langle P_{\beta} \right\rangle} \right)} \right\rangle}}} \leq {h_{{ma}\; x}^{2}\left( {T + {As}_{{ma}\; x}^{2}} \right)}}},$

Where s_(max)=max_(i)|s_(i)| is the number of elements in the largestTPB set. Keeping the same total number of measurements TS as in theabove equation for the error, the error on the mean in this case, usingthe grouping, is given by

${\epsilon = {\sqrt{\frac{{Var}^{G}\lbrack H\rbrack}{S}} \leq \sqrt{\frac{A\;{h_{{ma}\; x}^{2}\left( {T + {As}_{{ma}\; x}^{2}} \right)}}{TS}}}},$which can be compared to the case in which one samples the single Pauliterms individually (above). The error contribution from the covariance(which can be positive or negative) has to be traded off against the useof less samples from grouping. Further, the energy and Hamiltonianvariance can be estimated as

${= {\frac{1}{S}{\sum\limits_{i = 1}^{S}X_{i,\alpha}}}},{= {\sum\limits_{i = 1}^{A}{\sum\limits_{\alpha,{\beta \in s_{i}}}{h_{\alpha}h_{\beta}{{cov}(,)}}}}},$where X_(i, α) is the outcome of the i-th measurement on the α-th Pauliterm. Further yet, the covariance matrix element is defined after Smeasurements as

cov ⁡ ( , ) = 1 S - 1 ⁢ ∑ i = 1 S ⁢ ( X i , α - k ) ⁢ ( X i , β - ) .

Referring back to the FIG. 6, the method further includes reading outthe qubit states, as shown at block 634. The qubits 215 are furtherreset, in preparation of a further iteration, if required, as shown atblock 635. The reset is performed using a microwave pulse only scheme,whereby the computer 100 sends a reset microwave pulse to the quantumcomputer 300, specifically for the qubit(s) that is to be reset. Thereset microwave pulse is a predetermined pulse associated with theresonators of the quantum computer 300, and may vary according to thequantum computer 300 used. The method loops over the number of TPB sets,as shown at block 630. The expectation value of each set of Paulioperators is obtained with the same trial state.

Further, the qubit states are used to compute energy using the S sampledqubit states, as shown at block 640. The method further includesdetermining if the N control updates have been completed yet, as shownat block 660. If N iterations have not been completed, the methodincludes using an optimizer to analyze the energy estimate values thathave been captured using the present trial state, and optimize the trialstate by revising the pulse sequences 410 being used for theentanglement, as shown at block 670. For example, the optimization maybe performed using machine learning algorithms, such as a gradientdescent algorithm like a simultaneous perturbation stochastic algorithm(SPSA). The optimization provides updated angles according to which thecontrol pulses (microwave) 401 are sent to the qubits 215.

The energy estimate before every update of the angles {right arrow over(θ)} is based on a number of parameters p=N(3d+2), and grows linearlywith the depth of the circuit d and the number of qubits Q. As thenumber of parameters increases the optimization component of thealgorithm increases in overheads. The accuracy of the optimization mayalso be significantly lowered by the presence of energy fluctuations atthe k-th step ε_(k). Furthermore, on the quantum hardware 300, there aretime overheads associated with loading of pulse waveforms on theelectronics, resonator and qubit reset, and repeated sampling of thequbit readout. The optimizer also has to be robust to statisticalfluctuations, and use the least number of energy measurements periteration. The SPSA gives a level of accuracy in the optimization of thecost function that is comparable with finite-difference gradientapproximations, while saving an order O(p) of cost function evaluationsand is applicable in the context of quantum control and quantumtomography.

In one or more examples, In the SPSA approach, for every step k of theoptimization, samples from p symmetrical Bernoulli distributions {rightarrow over (Δ_(k))} are used along with preassigned elements from twosequences converging to zero, c_(k) and a_(k). The gradient at {rightarrow over (θ_(k))} is approximated using energy evaluations at {rightarrow over (θ)}_(k) ^(±)={right arrow over (θ_(k))}±c_(k){right arrowover (Δ)}_(k), and is constructed as

${{{\overset{\rightarrow}{g}}_{k}\left( {\overset{\rightarrow}{\theta}}_{k} \right)} = {\frac{\left\langle {{\Phi\left( {\overset{\rightarrow}{\theta}}_{k}^{+} \right)}{H}{\Phi\left( {\overset{\rightarrow}{\theta}}_{k}^{+} \right)}} \right\rangle - \left\langle {{\Phi\left( {\overset{\rightarrow}{\theta}}_{k}^{-} \right)}{H}{\Phi\left( {\overset{\rightarrow}{\theta}}_{k}^{-} \right)}} \right\rangle}{2c_{k}}{\overset{\rightarrow}{\Delta}}_{k}}},$as illustrated in FIG. 7 in plot 710. The gradient approximation onlyrequires two estimations of the energy, regardless of the number p ofvariables in {right arrow over (θ)}. The controls are then updated as{right arrow over (θ_(k+1))}={right arrow over (θ_(k))}−a_(k){rightarrow over (g_(k))}({right arrow over (θ_(k))}).

The convergence of θ_(k) to the optimal solution {right arrow over (θ)}*happens even in the presence of stochastic fluctuations, if the startingpoint is in the domain of the attraction of the problem. Convergenceremains an open issue if the starting point for the controls is not in adomain of attraction. In this case strategies like multiple competingstarting points are adopted. For example, sequences c_(k), a_(k) arechosen as

${c_{k} = \frac{c}{k^{\gamma}}},{a_{k} = {\frac{a}{k^{\alpha}}.}}$

Here, the parameters α, γ are predetermined, for example {α, γ}={0.602,0:101}, for ensuring the smoothest descent along the approximategradients defined for {right arrow over (θ_(k+1))} above. The value of cis tuned to adjust the robustness of the gradient evaluation withrespect to the magnitude of the energy fluctuations. In one or moreexamples, large fluctuations of the energy require gradient evaluationswith large c_(k), so that the fluctuations do not substantially affectthe gradient approximation. This condition is valid in the regime|

Φ({right arrow over (θ)}_(k) ⁺)|H|Φ({right arrow over (θ)}_(k) ⁺)

−

Φ({right arrow over (θ)}_(k) ⁻)|H|Φ({right arrow over (θ)}_(k) ⁻)

|>>∈_(k>)as shown in 710. In other words, good gradient approximations areobtained if the energy difference is larger than the stochasticfluctuations on the energy, and the parameter c is heuristically chosento meet this condition. For example, c=10⁻¹ is used to ensure robustnessin situations that that include decoherence noise and energyfluctuations, while the smaller c=10⁻² factor is used in the numericaloptimizations where the energy is evaluated without fluctuations.

The parameter a is then calibrated experimentally in order to achieve areasonable angle update on the first step of the optimization, forexample as |θ₂ ^((i))−θ₁ ^((i))|=2π/10, for all the angles i=1, 2, . . .p. In one or more examples, to achieve this, an inverse formula is usedthat is based on

${a = {\frac{2\pi}{5}\frac{c}{\left\langle {{\left\langle {{\Phi\left( {\overset{\rightarrow}{\theta}}_{1}^{+} \right)}{H}{\Phi\left( {\overset{\rightarrow}{\theta}}_{1}^{+} \right)}} \right\rangle - \left\langle {{\Phi\left( {\overset{\rightarrow}{\theta}}_{1}^{-} \right)}{H}{\Phi\left( {\overset{\rightarrow}{\theta}}_{1}^{-} \right)}} \right\rangle}} \right\rangle_{{\overset{\rightarrow}{\Delta}}_{1}}}}},$

Where the notation

_({right arrow over (Δ)}) ₁ indicates an average over different samplesfrom the distribution {right arrow over (Δ)}₁ that generates the firstgradient approximation. In one or more examples, by averaging alongdifferent directions, measure the average slope of the functionallandscape

Φ({right arrow over (θ)}_(k))|H|Φ({right arrow over (θ)}_(k))

of can be measured in the vicinity of the starting point {right arrowover (θ)}₁ and calibrate the experiment accordingly.

For example, FIG. 7 includes example plots 720 and 730 depictingparameter a being calibrated by measuring 25 (times the energiesE({right arrow over (θ)}₁ ^(±))=

Φ({right arrow over (θ)}₁ ^(±))|H|Φ({right arrow over (θ)}₁ ^(±))

measured for the LiH molecule at the depicted bond distances, with d=1,from the starting angles {right arrow over (θ)}₁ for different randomgradients approximations. It should be noted that during theoptimization by the method execution, the value of the energy

Φ({right arrow over (θ)}_(k))|H|Φ({right arrow over (θ)}_(k))

for the k-th optimized angles are not measured and instead only thevalues

Φ({right arrow over (θ)}₁ ⁺)|H|Φ({right arrow over (θ)}₁ ⁺)

and

Φ({right arrow over (θ)}₁ ⁻)|H|Φ({right arrow over (θ)}₁ ⁻)

are measured and reported, which serve to generate a new gradientapproximation. The underlying optimized angles {right arrow over(θ)}_(k) are only measured at the end of the optimization, averagingover the last 25 {right arrow over (θ)}_(k) ⁺ and 25 {right arrow over(θ)}_(k) ⁻ to further minimize stochastic fluctuations effect.Further-more, this last average is done with 10⁵ samples, as opposed tothe 10³ samples used to generate {right arrow over (θ)}_(k) ⁺ and {rightarrow over (θ)}_(k) ⁻ during the optimization, in order to reduce theerror on the measurement.

Referring back to the flowchart of FIG. 6, once the N iterations withcontrol updates using the optimizer have been performed, and if theenergy convergence has not yet occurred, the parameters are incrementedby predetermined values, as shown at block 680. For example, the numberof samples S is increased, and the number of iterations N is increased.Further, in one or more examples, the depth d of the quantum circuit 400is increased. In one or more examples, the depth d is increased after Nhas been increased to a predetermined maximum value, or if N has beenincreased a predetermined number of times.

The energy

Φ({right arrow over (θ)}_(k))|H|Φ({right arrow over (θ)}_(k))

≡

H

_(k) for the k-th control is evaluated, as shown at block 685. Forexample, the computer 100 determines if the energy states areconverging. If the energy converges, the energy is reported as asolution of the VQE. For example, the energy state estimate being withina predetermined threshold of a predetermined expected value is deemed asthe energy estimates reaching convergence. For example, an energy errorof approximately 0.0016 Hartree may be used to deem convergence when thebest energy estimate is close to the exact solution up to chemicalaccuracy. If the convergence is detected, the execution of the methodmay be stopped with the trial state being used provided as the solution.For example, the trial state in the ongoing molecular structure exampleproviding angles and/distances between the particles of the atom beinganalyzed. If the energy estimate, which is the optimization outcome forthe feedback loop between the computer 100 and the quantum computer 300,has converged, in this case increasing, changing the parameters d, N,and/or S, will not improve the final answer for the energy estimate anyfurther. Hence, the execution of the method is deemed completed.

If the energy estimate has not converged, the method continues toiterate until resource saturation of the quantum computer 300 isdetected. Once the parameters have been increased, the method includesvalidating that the incremented values do not cause resource saturationat the quantum circuit 400, as shown at block 690. For example,predetermined decoherence limit, and/or a predetermined sampling time,may be used to limit how far the parameters are increased, because asthe parameters are increased, the decoherence and the sampling time forgenerating the qubit states associated with measuring the energy statesincrease.

If the saturation limits are not met, the method iterates through theoperations described herein until one or more of the stopping conditionsare met. In one or more examples, a calibration time is monitored andthe quantum computer 300 is recalibrated after every calibration timeduration.

FIG. 8 depicts an example optimization of molecular structureapplication using the improved VQE according to one or more embodiments.Plot 805 in FIG. 8 depicts parity mapping of 8 spin orbitals onto 8qubits, reduced to 6 qubits via qubit tapering of fermionic spin-paritysymmetries according to one or more embodiments. The bars indicate theparity of the spin-orbitals encoded in each qubit 215. For example, themapping may be used to map the 8 spin-orbital Hamiltonian of BeH2spin-orbital Hamiltonian using the parity mapping, and remove, as in thecase of H2, two qubits associated to the spin-parity symmetries,reducing this to a 6 qubit problem that encodes 8 spin-orbitals. Asimilar approach may also be used to map LiH onto 4 qubits. The mappingis based on the mapping of the Hamiltonians for H2, LiH and BeH2 attheir bond distance described earlier.

Further, the results from an optimization procedure are illustrated inthe plot 810 of FIG. 8, using the BeH2 Hamiltonian for the interatomicdistance of 1.7 Å. The results presented are for d=1, with a total of 30Euler control angles (p=30) associated with 6 qubits. The inset of plot810 shows the simultaneous perturbation of 30 Euler angles, as theenergy estimates are updated. In this example scenario, to obtain thepotential energy surfaces for H2, LiH, and BeH2, the ground state energyof their molecular Hamiltonians is searched using 2, 4, and 6 qubitsrespectively, for depth d=1, for a range of different interatomicdistances. Here, for each iteration k, the gradient at each control{right arrow over (θ)}_(k) is approximated using 10³ samples (S=10³) forenergy estimations at {right arrow over (θ)}_(k) ⁺ and {right arrow over(θ)}_(k) ⁻. The inset shows the simultaneous optimization of 30. Eulerangles that control the trial state preparation. The final energyestimate is obtained using the angles {right arrow over (θ)}_(final)averaged over the last predetermined number of angle updates, forexample last 25. The last predetermined number of angle updates are usedin one or more examples in order to mitigate the effect of stochasticfluctuations, with a higher number of 10⁵ samples, to get a moreaccurate energy estimation.

FIG. 9 depicts a comparison of results using the improved VQE using thetrial states according to one or more embodiments with known values. Theplots a, b, and c of FIG. 9 depict comparisons of the outcomes ofmolecular structure modeling for a number of interatomic distances forH2 (plot a) LiH (plot b), and BeH2 (plot c). The results from thequantum computer 300 are compared with outcomes from numericalsimulations using depth d=1 circuits. Further, in FIG. 9, the top insetsof each plot highlight the qubits used for each molecule modeling, andthe cross-resonance gates that constitute U_(ENT). For example, for H2molecule, cross-resonance (CR) gates between qubits Q2 and Q4 constitutethe entangler U_(ENT), for LiH molecule, CR gates between qubit pairsQ2-Q4, Q2-Q1, and Q1-Q3 constitute the entangler U_(ENT), and for theBeH2 molecule, CR gates between qubit pairs Q2-Q4, Q2-Q1, Q1-Q3, Q4-Q5,and Q6-Q5 constitute the entangler U_(ENT). The bottom insets of eachplot are representations of the molecular geometry, not drawn to scale.For all the three molecules, the deviation of the experimental resultsfrom the exact curves, is well explained by the stochastic simulations.

The technical solutions described herein thus facilitates use of ahardware-efficient trial state preparation that relies on gates that areaccessible on quantum hardware. In one or more examples, the technicalsolutions described herein Use a fixed frequency superconducting qubitarchitecture. For example, the trial state preparation may use offixed-frequency single-qubit and entangling quantum gates for trialstate preparation, implemented using solely microwave pulses. Themicrowave pulses can be parameterized by variable amplitude and phase.Alternatively, the microwave pulses can be parameterized by constantamplitude with varying phase. The technical solutions described hereinfurther facilitate use of parallelization of quantum gates for trialstate preparation. For example, the technical solutions described hereindescribe the use of a unique set of single-qubit post-rotations at thedifferent qubits in the quantum hardware, the rotations implementedusing microwave pulses, for a measurement of multiple Hamiltonian termsthat belong to the same tensor product basis set. The technicalsolutions described herein further facilitate use of a microwave-onlyqubit reset scheme.

Thus, the technical solutions described herein improve the use of VQEfor quantum simulation, and thus provide an improvement to computingtechnology itself by generating and using hardware efficient trialstates based on a superconducting quantum processor. The technicalsolutions described herein further improve the application of VQE forquantum simulation because the technical solutions described herein areimplementable on near-term quantum hardware as opposed to typicalsolutions including full parameterization of unitaries in a UCC.Further, the technical solutions described herein facilitate mappingFermions to qubits in an efficient manner for simulation of theHamiltonians, (for example mapping 8 Hamiltonians to 6 qubits), thusimproving efficiency of the quantum computer used. The efficiency isfurther improved by using a measurement scheme by binning Paulioperators diagonal in a Tensor Product Basis (TPB).

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on the user's computer, partly on the user's computer, as astand-alone software package, partly on the user's computer and partlyon a remote computer or entirely on the remote computer or server. Inthe latter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

What is claimed is:
 1. A system comprising: a memory device includingcomputer-executable instructions; and a processor coupled with thememory and configured to execute the computer-executable instructions togenerate trial states for a variational quantum Eigenvalue solver (VQE)using a quantum computer that comprises a plurality of qubits, thegeneration comprising: selecting a number of samples S to capture fromthe qubits for a particular trial state, the samples comprisingmeasurements of qubit states; mapping a Hamiltonian to the qubits of thequantum computer; setting up an entangler in the quantum computer, theentangler defining an entangling interaction between at least a subsetof the qubits of the quantum computer, using an interleaved sequence ofarbitrary single-qubit rotations and entangler operations; reading out,from the quantum computer, the qubit states after post-rotationsassociated with Pauli terms in the Hamiltonian, the qubit statesrepresenting expectation values of the Pauli terms, and the reading outbeing performed for the selected number of samples S; computing anestimated energy state using the measured expectation values of thePauli terms in the Hamiltonian; and in response to the estimated energystate not converging with an expected energy state, computing a newtrial state for the VQE and iterating to compute the estimated energyusing the new trial state.
 2. The system of claim 1, wherein thegeneration of the trial states further comprises: selecting a number ofiterations N for which to update a trial state for the VQE; computingthe new trial state for the VQE and iterating to compute the estimatedenergy using the new trial state for the selected N iterations; and inresponse to the estimated energy state not converging with the expectedenergy state after the selected N iterations, incrementing the selectednumber of samples S and the selected number of iterations N.
 3. Thesystem of claim 2, wherein the generation of the trial states furthercomprises: in response to the estimated energy state not converging withthe expected energy state after the selected N iterations, incrementinga depth d associated with the quantum computer, the depth d indicating anumber of entanglers used to setup the trial state.
 4. The system ofclaim 1, wherein the entangler is a sequence of gates that entangle thequbits in the quantum computer.
 5. The system of claim 4, wherein thegates in the entangler and in the single-qubit rotations operate usingparallelization for trial state preparation.
 6. The system of claim 1,wherein a unique set of single-qubit rotations, implemented usingmicrowave pulses, is used for measurement of multiple Hamiltonian termsthat belong to the same tensor product basis set.
 7. The system of claim1, wherein the generation of the trial states further comprises:resetting the qubits of the quantum computer after each read out of thequbit states, the resetting comprising sending a reset pulse to thequantum computer.
 8. The system of claim 1, the trial states beingprepared using microwave pulses that are parameterized by waveform,amplitude, and time.
 9. A method for to generating trial states for avariational quantum Eigenvalue solver (VQE) using a quantum computerthat comprises a plurality of qubits, the generation comprising:selecting a number of samples S to capture from the qubits for aparticular trial state, the samples comprising measurements of qubitstates; mapping a Hamiltonian to the qubits of the quantum computer;setting up an entangler in the quantum computer, the entangler definingan entangling interaction between at least a subset of the qubits of thequantum computer, the setting up using an interleaved sequence ofarbitrary single qubit rotations and entangler operations; reading out,from the quantum computer, the measurements of qubit states afterpost-rotations associated with Pauli terms, the reading out beingperformed for the selected number of samples S; computing an estimatedenergy state using the measurements of the Pauli terms; and in responseto the estimated energy state not converging with an expected energystate, computing a new trial state for the VQE and iterating to computethe estimated energy using the new trial state.
 10. The method of claim9, further comprising: selecting a number of iterations N for which toupdate a trial state for the VQE; computing the new trial state for theVQE and iterating to compute the estimated energy using the new trialstate for the selected N iterations; and in response to the estimatedenergy state not converging with the expected energy state after theselected N iterations, incrementing the selected number of samples S andthe selected number of iterations N.
 11. The method of claim 10, furthercomprising, in response to the estimated energy state not convergingwith the expected energy state after the selected N iterations,incrementing a depth d associated with the quantum computer, the depth dindicating a number of entanglers in the trial state.
 12. The method ofclaim 9, wherein the entangler is a sequence of gates that entangle thequbits in the quantum computer.
 13. The method of claim 12, wherein thegates in the entangler operate using parallelization for trial statepreparation.
 14. The method of claim 9, wherein a unique set ofsingle-qubit post-rotations, implemented using microwave pulses, is usedfor measurement of multiple Hamiltonian terms that belong to the sametensor product basis set.
 15. The method of claim 9, further comprising:resetting the qubits of the quantum computer after each read out of thequbit states, the resetting comprising sending a reset pulse to thequantum computer; and initializing the qubits to a ground state prior toeach post-rotation.
 16. A quantum computing device comprising: aplurality of qubits; a plurality of resonators corresponding to each ofthe qubits, each resonator configured to receive control signals for acorresponding qubit, and to send readout signal representing a qubitstate of the corresponding qubit; wherein the quantum computing devicegenerates trial states for a variational quantum Eigenvalue solver(VQE), the generation comprising: receiving, by the resonators, controlpulses that prepare a trial state; setting up an entangler, theentangler defining an entangling interaction between at least a subsetof the qubits, using an interleaved sequence of arbitrary single qubitrotations and entangler operations in order to prepare the trial state;reading out, by the resonators, qubit states after post-rotationsassociated with Pauli terms in a Hamiltonian, the reading out beingperformed for a selected number of samples S for computing an estimatedenergy state; and in response to the estimated energy state notconverging with an expected energy state, receiving control signals toupdate to a new trial state for the VQE.
 17. The quantum computingdevice of claim 16, wherein in response to the estimated energy statenot converging with the expected energy state after a selected number ofN iterations, each iteration updating the trial state for the VQE:incrementing the selected number of samples S and the selected number ofiterations N.
 18. The quantum computing device of claim 16, wherein inresponse to the estimated energy state not converging with the expectedenergy state after the selected N iterations, incrementing a depth dassociated with the quantum computer, the depth d indicating a number ofentanglers in the trial state.
 19. The quantum computing device of claim16, wherein each of the resonators is configured to receive a resetpulse for resetting the corresponding qubit.
 20. The quantum computingdevice of claim 16, wherein gates in the entangler operate usingparallelization for trial state preparation.